LMFDB - Embedded newform 7440.2.a.bn.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-3,0,2,0,3,0,-2,0,2,0,3,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -0.395932 q^{7} +1.00000 q^{9} -2.79186 q^{11} +5.70682 q^{13} +1.00000 q^{15} +4.25951 q^{17} +6.10275 q^{19} +0.395932 q^{21} +6.36226 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.96633 q^{29} -1.00000 q^{31} +2.79186 q^{33} +0.395932 q^{35} -9.80957 q^{37} -5.70682 q^{39} +3.20814 q^{41} +3.31088 q^{43} -1.00000 q^{45} -7.05137 q^{47} -6.84324 q^{49} -4.25951 q^{51} -1.05137 q^{53} +2.79186 q^{55} -6.10275 q^{57} -3.34456 q^{59} +12.6218 q^{61} -0.395932 q^{63} -5.70682 q^{65} -13.8096 q^{67} -6.36226 q^{69} -13.5501 q^{71} +13.8096 q^{73} -1.00000 q^{75} +1.10539 q^{77} +0.156762 q^{79} +1.00000 q^{81} -7.74049 q^{83} -4.25951 q^{85} -7.96633 q^{87} +7.55005 q^{89} -2.25951 q^{91} +1.00000 q^{93} -6.10275 q^{95} +4.79186 q^{97} -2.79186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{15} + 10 q^{17} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 16 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 3 * q^15 + 10 * q^17 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 16 * q^41 - 2 * q^43 - 3 * q^45 - 12 * q^47 - 5 * q^49 - 10 * q^51 + 6 * q^53 + 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^63 - 2 * q^65 - 8 * q^67 + 2 * q^69 - 10 * q^71 + 8 * q^73 - 3 * q^75 + 28 * q^77 + 16 * q^79 + 3 * q^81 - 26 * q^83 - 10 * q^85 - 6 * q^87 - 8 * q^89 - 4 * q^91 + 3 * q^93 + 8 * q^97 - 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.395932	−0.149648	−0.0748241	−	0.997197i	$-0.523840\pi$
$7$	−0.395932	−0.149648	−0.0748241	+	0.997197i	$0.523840\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.79186	−0.841779	−0.420889	−	0.907112i	$-0.638282\pi$
$11$	−2.79186	−0.841779	−0.420889	+	0.907112i	$0.638282\pi$
$12$	0	0
$13$	5.70682	1.58279	0.791393	−	0.611308i	$-0.209356\pi$
$13$	5.70682	1.58279	0.791393	+	0.611308i	$0.209356\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.25951	1.03308	0.516542	−	0.856262i	$-0.327219\pi$
$17$	4.25951	1.03308	0.516542	+	0.856262i	$0.327219\pi$
$18$	0	0
$19$	6.10275	1.40007	0.700033	−	0.714110i	$-0.253168\pi$
$19$	6.10275	1.40007	0.700033	+	0.714110i	$0.253168\pi$
$20$	0	0
$21$	0.395932	0.0863994
$22$	0	0
$23$	6.36226	1.32662	0.663311	−	0.748344i	$-0.269151\pi$
$23$	6.36226	1.32662	0.663311	+	0.748344i	$0.269151\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.96633	1.47931	0.739655	−	0.672986i	$-0.234989\pi$
$29$	7.96633	1.47931	0.739655	+	0.672986i	$0.234989\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.79186	0.486001
$34$	0	0
$35$	0.395932	0.0669247
$36$	0	0
$37$	−9.80957	−1.61268	−0.806341	−	0.591451i	$-0.798555\pi$
$37$	−9.80957	−1.61268	−0.806341	+	0.591451i	$0.798555\pi$
$38$	0	0
$39$	−5.70682	−0.913822
$40$	0	0
$41$	3.20814	0.501027	0.250513	−	0.968113i	$-0.419401\pi$
$41$	3.20814	0.501027	0.250513	+	0.968113i	$0.419401\pi$
$42$	0	0
$43$	3.31088	0.504905	0.252453	−	0.967609i	$-0.418763\pi$
$43$	3.31088	0.504905	0.252453	+	0.967609i	$0.418763\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−7.05137	−1.02855	−0.514274	−	0.857626i	$-0.671939\pi$
$47$	−7.05137	−1.02855	−0.514274	+	0.857626i	$0.671939\pi$
$48$	0	0
$49$	−6.84324	−0.977605
$50$	0	0
$51$	−4.25951	−0.596451
$52$	0	0
$53$	−1.05137	−0.144417	−0.0722087	−	0.997390i	$-0.523005\pi$
$53$	−1.05137	−0.144417	−0.0722087	+	0.997390i	$0.523005\pi$
$54$	0	0
$55$	2.79186	0.376455
$56$	0	0
$57$	−6.10275	−0.808329
$58$	0	0
$59$	−3.34456	−0.435424	−0.217712	−	0.976013i	$-0.569859\pi$
$59$	−3.34456	−0.435424	−0.217712	+	0.976013i	$0.569859\pi$
$60$	0	0
$61$	12.6218	1.61605	0.808026	−	0.589147i	$-0.200536\pi$
$61$	12.6218	1.61605	0.808026	+	0.589147i	$0.200536\pi$
$62$	0	0
$63$	−0.395932	−0.0498827
$64$	0	0
$65$	−5.70682	−0.707843
$66$	0	0
$67$	−13.8096	−1.68711	−0.843553	−	0.537045i	$-0.819540\pi$
$67$	−13.8096	−1.68711	−0.843553	+	0.537045i	$0.819540\pi$
$68$	0	0
$69$	−6.36226	−0.765926
$70$	0	0
$71$	−13.5501	−1.60810	−0.804048	−	0.594565i	$-0.797324\pi$
$71$	−13.5501	−1.60810	−0.804048	+	0.594565i	$0.797324\pi$
$72$	0	0
$73$	13.8096	1.61629	0.808144	−	0.588985i	$-0.200472\pi$
$73$	13.8096	1.61629	0.808144	+	0.588985i	$0.200472\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.10539	0.125971
$78$	0	0
$79$	0.156762	0.0176371	0.00881855	−	0.999961i	$-0.497193\pi$
$79$	0.156762	0.0176371	0.00881855	+	0.999961i	$0.497193\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.74049	−0.849629	−0.424815	−	0.905280i	$-0.639661\pi$
$83$	−7.74049	−0.849629	−0.424815	+	0.905280i	$0.639661\pi$
$84$	0	0
$85$	−4.25951	−0.462009
$86$	0	0
$87$	−7.96633	−0.854080
$88$	0	0
$89$	7.55005	0.800304	0.400152	−	0.916449i	$-0.368957\pi$
$89$	7.55005	0.800304	0.400152	+	0.916449i	$0.368957\pi$
$90$	0	0
$91$	−2.25951	−0.236861
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−6.10275	−0.626129
$96$	0	0
$97$	4.79186	0.486540	0.243270	−	0.969959i	$-0.421780\pi$
$97$	4.79186	0.486540	0.243270	+	0.969959i	$0.421780\pi$
$98$	0	0
$99$	−2.79186	−0.280593
$100$	0	0
$101$	0.272843	0.0271489	0.0135744	−	0.999908i	$-0.495679\pi$
$101$	0.272843	0.0271489	0.0135744	+	0.999908i	$0.495679\pi$
$102$	0	0
$103$	2.66877	0.262962	0.131481	−	0.991319i	$-0.458027\pi$
$103$	2.66877	0.262962	0.131481	+	0.991319i	$0.458027\pi$
$104$	0	0
$105$	−0.395932	−0.0386390
$106$	0	0
$107$	−10.2595	−0.991824	−0.495912	−	0.868373i	$-0.665166\pi$
$107$	−10.2595	−0.991824	−0.495912	+	0.868373i	$0.665166\pi$
$108$	0	0
$109$	−13.5704	−1.29981	−0.649904	−	0.760016i	$-0.725191\pi$
$109$	−13.5704	−1.29981	−0.649904	+	0.760016i	$0.725191\pi$
$110$	0	0
$111$	9.80957	0.931083
$112$	0	0
$113$	−9.93265	−0.934386	−0.467193	−	0.884155i	$-0.654735\pi$
$113$	−9.93265	−0.934386	−0.467193	+	0.884155i	$0.654735\pi$
$114$	0	0
$115$	−6.36226	−0.593284
$116$	0	0
$117$	5.70682	0.527595
$118$	0	0
$119$	−1.68648	−0.154599
$120$	0	0
$121$	−3.20550	−0.291409
$122$	0	0
$123$	−3.20814	−0.289268
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	17.8299	1.58215	0.791074	−	0.611720i	$-0.209522\pi$
$127$	17.8299	1.58215	0.791074	+	0.611720i	$0.209522\pi$
$128$	0	0
$129$	−3.31088	−0.291507
$130$	0	0
$131$	14.9637	1.30738	0.653692	−	0.756761i	$-0.273219\pi$
$131$	14.9637	1.30738	0.653692	+	0.756761i	$0.273219\pi$
$132$	0	0
$133$	−2.41627	−0.209517
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	14.8813	1.27139	0.635697	−	0.771939i	$-0.280713\pi$
$137$	14.8813	1.27139	0.635697	+	0.771939i	$0.280713\pi$
$138$	0	0
$139$	11.4136	0.968092	0.484046	−	0.875043i	$-0.339167\pi$
$139$	11.4136	0.968092	0.484046	+	0.875043i	$0.339167\pi$
$140$	0	0
$141$	7.05137	0.593833
$142$	0	0
$143$	−15.9327	−1.33236
$144$	0	0
$145$	−7.96633	−0.661567
$146$	0	0
$147$	6.84324	0.564421
$148$	0	0
$149$	−8.79186	−0.720258	−0.360129	−	0.932903i	$-0.617267\pi$
$149$	−8.79186	−0.720258	−0.360129	+	0.932903i	$0.617267\pi$
$150$	0	0
$151$	23.7759	1.93485	0.967427	−	0.253149i	$-0.0814662\pi$
$151$	23.7759	1.93485	0.967427	+	0.253149i	$0.0814662\pi$
$152$	0	0
$153$	4.25951	0.344361
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	17.5164	1.39796	0.698980	−	0.715141i	$-0.253638\pi$
$157$	17.5164	1.39796	0.698980	+	0.715141i	$0.253638\pi$
$158$	0	0
$159$	1.05137	0.0833794
$160$	0	0
$161$	−2.51902	−0.198527
$162$	0	0
$163$	16.4987	1.29228	0.646138	−	0.763220i	$-0.276383\pi$
$163$	16.4987	1.29228	0.646138	+	0.763220i	$0.276383\pi$
$164$	0	0
$165$	−2.79186	−0.217346
$166$	0	0
$167$	10.8946	0.843050	0.421525	−	0.906817i	$-0.361495\pi$
$167$	10.8946	0.843050	0.421525	+	0.906817i	$0.361495\pi$
$168$	0	0
$169$	19.5678	1.50521
$170$	0	0
$171$	6.10275	0.466689
$172$	0	0
$173$	−16.8946	−1.28447	−0.642237	−	0.766506i	$-0.721993\pi$
$173$	−16.8946	−1.28447	−0.642237	+	0.766506i	$0.721993\pi$
$174$	0	0
$175$	−0.395932	−0.0299296
$176$	0	0
$177$	3.34456	0.251392
$178$	0	0
$179$	−2.93529	−0.219394	−0.109697	−	0.993965i	$-0.534988\pi$
$179$	−2.93529	−0.219394	−0.109697	+	0.993965i	$0.534988\pi$
$180$	0	0
$181$	7.03804	0.523134	0.261567	−	0.965185i	$-0.415761\pi$
$181$	7.03804	0.523134	0.261567	+	0.965185i	$0.415761\pi$
$182$	0	0
$183$	−12.6218	−0.933028
$184$	0	0
$185$	9.80957	0.721214
$186$	0	0
$187$	−11.8920	−0.869627
$188$	0	0
$189$	0.395932	0.0287998
$190$	0	0
$191$	−12.8609	−0.930585	−0.465292	−	0.885157i	$-0.654051\pi$
$191$	−12.8609	−0.930585	−0.465292	+	0.885157i	$0.654051\pi$
$192$	0	0
$193$	23.9327	1.72271	0.861355	−	0.508003i	$-0.169616\pi$
$193$	23.9327	1.72271	0.861355	+	0.508003i	$0.169616\pi$
$194$	0	0
$195$	5.70682	0.408674
$196$	0	0
$197$	−19.0868	−1.35988	−0.679938	−	0.733269i	$-0.737993\pi$
$197$	−19.0868	−1.35988	−0.679938	+	0.733269i	$0.737993\pi$
$198$	0	0
$199$	14.5324	1.03017	0.515086	−	0.857139i	$-0.327760\pi$
$199$	14.5324	1.03017	0.515086	+	0.857139i	$0.327760\pi$
$200$	0	0
$201$	13.8096	0.974052
$202$	0	0
$203$	−3.15412	−0.221376
$204$	0	0
$205$	−3.20814	−0.224066
$206$	0	0
$207$	6.36226	0.442208
$208$	0	0
$209$	−17.0380	−1.17855
$210$	0	0
$211$	−6.89461	−0.474645	−0.237322	−	0.971431i	$-0.576270\pi$
$211$	−6.89461	−0.474645	−0.237322	+	0.971431i	$0.576270\pi$
$212$	0	0
$213$	13.5501	0.928434
$214$	0	0
$215$	−3.31088	−0.225800
$216$	0	0
$217$	0.395932	0.0268776
$218$	0	0
$219$	−13.8096	−0.933164
$220$	0	0
$221$	24.3082	1.63515
$222$	0	0
$223$	16.3082	1.09208	0.546040	−	0.837759i	$-0.316134\pi$
$223$	16.3082	1.09208	0.546040	+	0.837759i	$0.316134\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	5.94599	0.394649	0.197324	−	0.980338i	$-0.436775\pi$
$227$	5.94599	0.394649	0.197324	+	0.980338i	$0.436775\pi$
$228$	0	0
$229$	−12.3756	−0.817802	−0.408901	−	0.912579i	$-0.634088\pi$
$229$	−12.3756	−0.817802	−0.408901	+	0.912579i	$0.634088\pi$
$230$	0	0
$231$	−1.10539	−0.0727292
$232$	0	0
$233$	19.7759	1.29556	0.647781	−	0.761827i	$-0.275697\pi$
$233$	19.7759	1.29556	0.647781	+	0.761827i	$0.275697\pi$
$234$	0	0
$235$	7.05137	0.459981
$236$	0	0
$237$	−0.156762	−0.0101828
$238$	0	0
$239$	29.3463	1.89825	0.949127	−	0.314894i	$-0.101969\pi$
$239$	29.3463	1.89825	0.949127	+	0.314894i	$0.101969\pi$
$240$	0	0
$241$	−13.9327	−0.897481	−0.448741	−	0.893662i	$-0.648127\pi$
$241$	−13.9327	−0.897481	−0.448741	+	0.893662i	$0.648127\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.84324	0.437198
$246$	0	0
$247$	34.8273	2.21601
$248$	0	0
$249$	7.74049	0.490534
$250$	0	0
$251$	7.68648	0.485166	0.242583	−	0.970131i	$-0.422005\pi$
$251$	7.68648	0.485166	0.242583	+	0.970131i	$0.422005\pi$
$252$	0	0
$253$	−17.7626	−1.11672
$254$	0	0
$255$	4.25951	0.266741
$256$	0	0
$257$	−13.3596	−0.833350	−0.416675	−	0.909055i	$-0.636805\pi$
$257$	−13.3596	−0.833350	−0.416675	+	0.909055i	$0.636805\pi$
$258$	0	0
$259$	3.88392	0.241335
$260$	0	0
$261$	7.96633	0.493103
$262$	0	0
$263$	−13.8299	−0.852789	−0.426394	−	0.904537i	$-0.640216\pi$
$263$	−13.8299	−0.852789	−0.426394	+	0.904537i	$0.640216\pi$
$264$	0	0
$265$	1.05137	0.0645854
$266$	0	0
$267$	−7.55005	−0.462056
$268$	0	0
$269$	13.0044	0.792890	0.396445	−	0.918058i	$-0.370244\pi$
$269$	13.0044	0.792890	0.396445	+	0.918058i	$0.370244\pi$
$270$	0	0
$271$	−7.61913	−0.462829	−0.231415	−	0.972855i	$-0.574335\pi$
$271$	−7.61913	−0.462829	−0.231415	+	0.972855i	$0.574335\pi$
$272$	0	0
$273$	2.25951	0.136752
$274$	0	0
$275$	−2.79186	−0.168356
$276$	0	0
$277$	−17.2232	−1.03484	−0.517421	−	0.855731i	$-0.673108\pi$
$277$	−17.2232	−1.03484	−0.517421	+	0.855731i	$0.673108\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−17.6191	−1.05107	−0.525535	−	0.850772i	$-0.676135\pi$
$281$	−17.6191	−1.05107	−0.525535	+	0.850772i	$0.676135\pi$
$282$	0	0
$283$	−8.08241	−0.480449	−0.240225	−	0.970717i	$-0.577221\pi$
$283$	−8.08241	−0.480449	−0.240225	+	0.970717i	$0.577221\pi$
$284$	0	0
$285$	6.10275	0.361496
$286$	0	0
$287$	−1.27020	−0.0749777
$288$	0	0
$289$	1.14343	0.0672606
$290$	0	0
$291$	−4.79186	−0.280904
$292$	0	0
$293$	29.6731	1.73352	0.866762	−	0.498722i	$-0.166197\pi$
$293$	29.6731	1.73352	0.866762	+	0.498722i	$0.166197\pi$
$294$	0	0
$295$	3.34456	0.194728
$296$	0	0
$297$	2.79186	0.162000
$298$	0	0
$299$	36.3082	2.09976
$300$	0	0
$301$	−1.31088	−0.0755581
$302$	0	0
$303$	−0.272843	−0.0156744
$304$	0	0
$305$	−12.6218	−0.722720
$306$	0	0
$307$	13.9797	0.797861	0.398931	−	0.916981i	$-0.369381\pi$
$307$	13.9797	0.797861	0.398931	+	0.916981i	$0.369381\pi$
$308$	0	0
$309$	−2.66877	−0.151821
$310$	0	0
$311$	−32.4180	−1.83826	−0.919128	−	0.393959i	$-0.871105\pi$
$311$	−32.4180	−1.83826	−0.919128	+	0.393959i	$0.871105\pi$
$312$	0	0
$313$	−22.1178	−1.25017	−0.625086	−	0.780556i	$-0.714936\pi$
$313$	−22.1178	−1.25017	−0.625086	+	0.780556i	$0.714936\pi$
$314$	0	0
$315$	0.395932	0.0223082
$316$	0	0
$317$	25.3596	1.42434	0.712169	−	0.702008i	$-0.247713\pi$
$317$	25.3596	1.42434	0.712169	+	0.702008i	$0.247713\pi$
$318$	0	0
$319$	−22.2409	−1.24525
$320$	0	0
$321$	10.2595	0.572630
$322$	0	0
$323$	25.9947	1.44638
$324$	0	0
$325$	5.70682	0.316557
$326$	0	0
$327$	13.5704	0.750445
$328$	0	0
$329$	2.79186	0.153920
$330$	0	0
$331$	−25.9460	−1.42612	−0.713060	−	0.701103i	$-0.752691\pi$
$331$	−25.9460	−1.42612	−0.713060	+	0.701103i	$0.752691\pi$
$332$	0	0
$333$	−9.80957	−0.537561
$334$	0	0
$335$	13.8096	0.754497
$336$	0	0
$337$	0.812204	0.0442436	0.0221218	−	0.999755i	$-0.492958\pi$
$337$	0.812204	0.0442436	0.0221218	+	0.999755i	$0.492958\pi$
$338$	0	0
$339$	9.93265	0.539468
$340$	0	0
$341$	2.79186	0.151188
$342$	0	0
$343$	5.48098	0.295945
$344$	0	0
$345$	6.36226	0.342533
$346$	0	0
$347$	−32.1382	−1.72527	−0.862633	−	0.505830i	$-0.831186\pi$
$347$	−32.1382	−1.72527	−0.862633	+	0.505830i	$0.831186\pi$
$348$	0	0
$349$	−21.2569	−1.13785	−0.568927	−	0.822388i	$-0.692641\pi$
$349$	−21.2569	−1.13785	−0.568927	+	0.822388i	$0.692641\pi$
$350$	0	0
$351$	−5.70682	−0.304607
$352$	0	0
$353$	2.88128	0.153355	0.0766775	−	0.997056i	$-0.475569\pi$
$353$	2.88128	0.153355	0.0766775	+	0.997056i	$0.475569\pi$
$354$	0	0
$355$	13.5501	0.719162
$356$	0	0
$357$	1.68648	0.0892578
$358$	0	0
$359$	28.6148	1.51023	0.755115	−	0.655593i	$-0.227581\pi$
$359$	28.6148	1.51023	0.755115	+	0.655593i	$0.227581\pi$
$360$	0	0
$361$	18.2435	0.960186
$362$	0	0
$363$	3.20550	0.168245
$364$	0	0
$365$	−13.8096	−0.722826
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	3.20814	0.167009
$370$	0	0
$371$	0.416273	0.0216118
$372$	0	0
$373$	−6.13815	−0.317821	−0.158911	−	0.987293i	$-0.550798\pi$
$373$	−6.13815	−0.317821	−0.158911	+	0.987293i	$0.550798\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	45.4624	2.34143
$378$	0	0
$379$	−2.30825	−0.118567	−0.0592833	−	0.998241i	$-0.518882\pi$
$379$	−2.30825	−0.118567	−0.0592833	+	0.998241i	$0.518882\pi$
$380$	0	0
$381$	−17.8299	−0.913454
$382$	0	0
$383$	−6.90794	−0.352979	−0.176490	−	0.984302i	$-0.556474\pi$
$383$	−6.90794	−0.352979	−0.176490	+	0.984302i	$0.556474\pi$
$384$	0	0
$385$	−1.10539	−0.0563358
$386$	0	0
$387$	3.31088	0.168302
$388$	0	0
$389$	−29.1745	−1.47920	−0.739602	−	0.673044i	$-0.764986\pi$
$389$	−29.1745	−1.47920	−0.739602	+	0.673044i	$0.764986\pi$
$390$	0	0
$391$	27.1001	1.37051
$392$	0	0
$393$	−14.9637	−0.754818
$394$	0	0
$395$	−0.156762	−0.00788755
$396$	0	0
$397$	19.7626	0.991854	0.495927	−	0.868364i	$-0.334828\pi$
$397$	19.7626	0.991854	0.495927	+	0.868364i	$0.334828\pi$
$398$	0	0
$399$	2.41627	0.120965
$400$	0	0
$401$	−19.5501	−0.976283	−0.488142	−	0.872764i	$-0.662325\pi$
$401$	−19.5501	−0.976283	−0.488142	+	0.872764i	$0.662325\pi$
$402$	0	0
$403$	−5.70682	−0.284277
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	27.3870	1.35752
$408$	0	0
$409$	−9.45431	−0.467486	−0.233743	−	0.972298i	$-0.575097\pi$
$409$	−9.45431	−0.467486	−0.233743	+	0.972298i	$0.575097\pi$
$410$	0	0
$411$	−14.8813	−0.734039
$412$	0	0
$413$	1.32422	0.0651605
$414$	0	0
$415$	7.74049	0.379966
$416$	0	0
$417$	−11.4136	−0.558928
$418$	0	0
$419$	6.40926	0.313113	0.156557	−	0.987669i	$-0.449961\pi$
$419$	6.40926	0.313113	0.156557	+	0.987669i	$0.449961\pi$
$420$	0	0
$421$	−4.05401	−0.197581	−0.0987903	−	0.995108i	$-0.531497\pi$
$421$	−4.05401	−0.197581	−0.0987903	+	0.995108i	$0.531497\pi$
$422$	0	0
$423$	−7.05137	−0.342850
$424$	0	0
$425$	4.25951	0.206617
$426$	0	0
$427$	−4.99736	−0.241839
$428$	0	0
$429$	15.9327	0.769236
$430$	0	0
$431$	9.52339	0.458726	0.229363	−	0.973341i	$-0.426336\pi$
$431$	9.52339	0.458726	0.229363	+	0.973341i	$0.426336\pi$
$432$	0	0
$433$	30.0151	1.44243	0.721216	−	0.692710i	$-0.243584\pi$
$433$	30.0151	1.44243	0.721216	+	0.692710i	$0.243584\pi$
$434$	0	0
$435$	7.96633	0.381956
$436$	0	0
$437$	38.8273	1.85736
$438$	0	0
$439$	31.3730	1.49735	0.748675	−	0.662938i	$-0.230691\pi$
$439$	31.3730	1.49735	0.748675	+	0.662938i	$0.230691\pi$
$440$	0	0
$441$	−6.84324	−0.325868
$442$	0	0
$443$	−18.4917	−0.878566	−0.439283	−	0.898349i	$-0.644767\pi$
$443$	−18.4917	−0.878566	−0.439283	+	0.898349i	$0.644767\pi$
$444$	0	0
$445$	−7.55005	−0.357907
$446$	0	0
$447$	8.79186	0.415841
$448$	0	0
$449$	6.44467	0.304143	0.152071	−	0.988370i	$-0.451406\pi$
$449$	6.44467	0.304143	0.152071	+	0.988370i	$0.451406\pi$
$450$	0	0
$451$	−8.95668	−0.421754
$452$	0	0
$453$	−23.7759	−1.11709
$454$	0	0
$455$	2.25951	0.105927
$456$	0	0
$457$	20.9504	0.980016	0.490008	−	0.871718i	$-0.336994\pi$
$457$	20.9504	0.980016	0.490008	+	0.871718i	$0.336994\pi$
$458$	0	0
$459$	−4.25951	−0.198817
$460$	0	0
$461$	11.8990	0.554191	0.277095	−	0.960842i	$-0.410628\pi$
$461$	11.8990	0.554191	0.277095	+	0.960842i	$0.410628\pi$
$462$	0	0
$463$	2.58637	0.120199	0.0600993	−	0.998192i	$-0.480858\pi$
$463$	2.58637	0.120199	0.0600993	+	0.998192i	$0.480858\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	7.05137	0.326299	0.163149	−	0.986601i	$-0.447835\pi$
$467$	7.05137	0.326299	0.163149	+	0.986601i	$0.447835\pi$
$468$	0	0
$469$	5.46765	0.252472
$470$	0	0
$471$	−17.5164	−0.807112
$472$	0	0
$473$	−9.24354	−0.425018
$474$	0	0
$475$	6.10275	0.280013
$476$	0	0
$477$	−1.05137	−0.0481391
$478$	0	0
$479$	32.9637	1.50615	0.753075	−	0.657935i	$-0.228570\pi$
$479$	32.9637	1.50615	0.753075	+	0.657935i	$0.228570\pi$
$480$	0	0
$481$	−55.9814	−2.55253
$482$	0	0
$483$	2.51902	0.114619
$484$	0	0
$485$	−4.79186	−0.217587
$486$	0	0
$487$	14.0621	0.637213	0.318607	−	0.947887i	$-0.396785\pi$
$487$	14.0621	0.637213	0.318607	+	0.947887i	$0.396785\pi$
$488$	0	0
$489$	−16.4987	−0.746096
$490$	0	0
$491$	−15.9327	−0.719031	−0.359515	−	0.933139i	$-0.617058\pi$
$491$	−15.9327	−0.719031	−0.359515	+	0.933139i	$0.617058\pi$
$492$	0	0
$493$	33.9327	1.52825
$494$	0	0
$495$	2.79186	0.125485
$496$	0	0
$497$	5.36490	0.240649
$498$	0	0
$499$	−21.9947	−0.984619	−0.492309	−	0.870420i	$-0.663847\pi$
$499$	−21.9947	−0.984619	−0.492309	+	0.870420i	$0.663847\pi$
$500$	0	0
$501$	−10.8946	−0.486735
$502$	0	0
$503$	−5.67314	−0.252953	−0.126476	−	0.991970i	$-0.540367\pi$
$503$	−5.67314	−0.252953	−0.126476	+	0.991970i	$0.540367\pi$
$504$	0	0
$505$	−0.272843	−0.0121413
$506$	0	0
$507$	−19.5678	−0.869035
$508$	0	0
$509$	−33.1071	−1.46745	−0.733724	−	0.679448i	$-0.762219\pi$
$509$	−33.1071	−1.46745	−0.733724	+	0.679448i	$0.762219\pi$
$510$	0	0
$511$	−5.46765	−0.241874
$512$	0	0
$513$	−6.10275	−0.269443
$514$	0	0
$515$	−2.66877	−0.117600
$516$	0	0
$517$	19.6865	0.865810
$518$	0	0
$519$	16.8946	0.741591
$520$	0	0
$521$	−23.6191	−1.03477	−0.517386	−	0.855752i	$-0.673095\pi$
$521$	−23.6191	−1.03477	−0.517386	+	0.855752i	$0.673095\pi$
$522$	0	0
$523$	0.0673457	0.00294482	0.00147241	−	0.999999i	$-0.499531\pi$
$523$	0.0673457	0.00294482	0.00147241	+	0.999999i	$0.499531\pi$
$524$	0	0
$525$	0.395932	0.0172799
$526$	0	0
$527$	−4.25951	−0.185547
$528$	0	0
$529$	17.4783	0.759928
$530$	0	0
$531$	−3.34456	−0.145141
$532$	0	0
$533$	18.3082	0.793018
$534$	0	0
$535$	10.2595	0.443557
$536$	0	0
$537$	2.93529	0.126667
$538$	0	0
$539$	19.1054	0.822927
$540$	0	0
$541$	−32.3303	−1.38999	−0.694994	−	0.719015i	$-0.744593\pi$
$541$	−32.3303	−1.38999	−0.694994	+	0.719015i	$0.744593\pi$
$542$	0	0
$543$	−7.03804	−0.302031
$544$	0	0
$545$	13.5704	0.581292
$546$	0	0
$547$	31.8716	1.36273	0.681366	−	0.731943i	$-0.261386\pi$
$547$	31.8716	1.36273	0.681366	+	0.731943i	$0.261386\pi$
$548$	0	0
$549$	12.6218	0.538684
$550$	0	0
$551$	48.6165	2.07113
$552$	0	0
$553$	−0.0620671	−0.00263936
$554$	0	0
$555$	−9.80957	−0.416393
$556$	0	0
$557$	−33.9140	−1.43698	−0.718492	−	0.695535i	$-0.755167\pi$
$557$	−33.9140	−1.43698	−0.718492	+	0.695535i	$0.755167\pi$
$558$	0	0
$559$	18.8946	0.799157
$560$	0	0
$561$	11.8920	0.502079
$562$	0	0
$563$	3.29755	0.138975	0.0694876	−	0.997583i	$-0.477864\pi$
$563$	3.29755	0.138975	0.0694876	+	0.997583i	$0.477864\pi$
$564$	0	0
$565$	9.93265	0.417870
$566$	0	0
$567$	−0.395932	−0.0166276
$568$	0	0
$569$	−17.8636	−0.748880	−0.374440	−	0.927251i	$-0.622165\pi$
$569$	−17.8636	−0.748880	−0.374440	+	0.927251i	$0.622165\pi$
$570$	0	0
$571$	10.0354	0.419969	0.209984	−	0.977705i	$-0.432659\pi$
$571$	10.0354	0.419969	0.209984	+	0.977705i	$0.432659\pi$
$572$	0	0
$573$	12.8609	0.537273
$574$	0	0
$575$	6.36226	0.265325
$576$	0	0
$577$	−30.8273	−1.28336	−0.641678	−	0.766974i	$-0.721761\pi$
$577$	−30.8273	−1.28336	−0.641678	+	0.766974i	$0.721761\pi$
$578$	0	0
$579$	−23.9327	−0.994607
$580$	0	0
$581$	3.06471	0.127145
$582$	0	0
$583$	2.93529	0.121567
$584$	0	0
$585$	−5.70682	−0.235948
$586$	0	0
$587$	−1.82991	−0.0755283	−0.0377641	−	0.999287i	$-0.512024\pi$
$587$	−1.82991	−0.0755283	−0.0377641	+	0.999287i	$0.512024\pi$
$588$	0	0
$589$	−6.10275	−0.251459
$590$	0	0
$591$	19.0868	0.785125
$592$	0	0
$593$	18.4517	0.757719	0.378860	−	0.925454i	$-0.376316\pi$
$593$	18.4517	0.757719	0.378860	+	0.925454i	$0.376316\pi$
$594$	0	0
$595$	1.68648	0.0691388
$596$	0	0
$597$	−14.5324	−0.594770
$598$	0	0
$599$	−28.6502	−1.17061	−0.585307	−	0.810812i	$-0.699026\pi$
$599$	−28.6502	−1.17061	−0.585307	+	0.810812i	$0.699026\pi$
$600$	0	0
$601$	14.4517	0.589496	0.294748	−	0.955575i	$-0.404764\pi$
$601$	14.4517	0.589496	0.294748	+	0.955575i	$0.404764\pi$
$602$	0	0
$603$	−13.8096	−0.562369
$604$	0	0
$605$	3.20550	0.130322
$606$	0	0
$607$	16.8796	0.685120	0.342560	−	0.939496i	$-0.388706\pi$
$607$	16.8796	0.685120	0.342560	+	0.939496i	$0.388706\pi$
$608$	0	0
$609$	3.15412	0.127812
$610$	0	0
$611$	−40.2409	−1.62797
$612$	0	0
$613$	36.8336	1.48769	0.743847	−	0.668350i	$-0.232999\pi$
$613$	36.8336	1.48769	0.743847	+	0.668350i	$0.232999\pi$
$614$	0	0
$615$	3.20814	0.129365
$616$	0	0
$617$	24.9620	1.00493	0.502465	−	0.864597i	$-0.332426\pi$
$617$	24.9620	1.00493	0.502465	+	0.864597i	$0.332426\pi$
$618$	0	0
$619$	47.3950	1.90497	0.952483	−	0.304591i	$-0.0985196\pi$
$619$	47.3950	1.90497	0.952483	+	0.304591i	$0.0985196\pi$
$620$	0	0
$621$	−6.36226	−0.255309
$622$	0	0
$623$	−2.98931	−0.119764
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	17.0380	0.680434
$628$	0	0
$629$	−41.7839	−1.66604
$630$	0	0
$631$	41.9947	1.67178	0.835892	−	0.548894i	$-0.184951\pi$
$631$	41.9947	1.67178	0.835892	+	0.548894i	$0.184951\pi$
$632$	0	0
$633$	6.89461	0.274036
$634$	0	0
$635$	−17.8299	−0.707558
$636$	0	0
$637$	−39.0531	−1.54734
$638$	0	0
$639$	−13.5501	−0.536032
$640$	0	0
$641$	−21.2365	−0.838793	−0.419396	−	0.907803i	$-0.637758\pi$
$641$	−21.2365	−0.838793	−0.419396	+	0.907803i	$0.637758\pi$
$642$	0	0
$643$	19.9327	0.786067	0.393034	−	0.919524i	$-0.371426\pi$
$643$	19.9327	0.786067	0.393034	+	0.919524i	$0.371426\pi$
$644$	0	0
$645$	3.31088	0.130366
$646$	0	0
$647$	29.4757	1.15881	0.579405	−	0.815040i	$-0.303285\pi$
$647$	29.4757	1.15881	0.579405	+	0.815040i	$0.303285\pi$
$648$	0	0
$649$	9.33755	0.366531
$650$	0	0
$651$	−0.395932	−0.0155178
$652$	0	0
$653$	12.8139	0.501448	0.250724	−	0.968059i	$-0.419331\pi$
$653$	12.8139	0.501448	0.250724	+	0.968059i	$0.419331\pi$
$654$	0	0
$655$	−14.9637	−0.584680
$656$	0	0
$657$	13.8096	0.538762
$658$	0	0
$659$	−26.7262	−1.04111	−0.520553	−	0.853829i	$-0.674274\pi$
$659$	−26.7262	−1.04111	−0.520553	+	0.853829i	$0.674274\pi$
$660$	0	0
$661$	17.9327	0.697499	0.348750	−	0.937216i	$-0.386606\pi$
$661$	17.9327	0.697499	0.348750	+	0.937216i	$0.386606\pi$
$662$	0	0
$663$	−24.3082	−0.944054
$664$	0	0
$665$	2.41627	0.0936990
$666$	0	0
$667$	50.6838	1.96249
$668$	0	0
$669$	−16.3082	−0.630513
$670$	0	0
$671$	−35.2383	−1.36036
$672$	0	0
$673$	45.2586	1.74459	0.872295	−	0.488979i	$-0.162631\pi$
$673$	45.2586	1.74459	0.872295	+	0.488979i	$0.162631\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	36.9433	1.41985	0.709924	−	0.704278i	$-0.248729\pi$
$677$	36.9433	1.41985	0.709924	+	0.704278i	$0.248729\pi$
$678$	0	0
$679$	−1.89725	−0.0728098
$680$	0	0
$681$	−5.94599	−0.227851
$682$	0	0
$683$	51.0815	1.95458	0.977290	−	0.211908i	$-0.0679677\pi$
$683$	51.0815	1.95458	0.977290	+	0.211908i	$0.0679677\pi$
$684$	0	0
$685$	−14.8813	−0.568584
$686$	0	0
$687$	12.3756	0.472158
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−43.6191	−1.65935	−0.829675	−	0.558247i	$-0.811474\pi$
$691$	−43.6191	−1.65935	−0.829675	+	0.558247i	$0.811474\pi$
$692$	0	0
$693$	1.10539	0.0419902
$694$	0	0
$695$	−11.4136	−0.432944
$696$	0	0
$697$	13.6651	0.517602
$698$	0	0
$699$	−19.7759	−0.747993
$700$	0	0
$701$	45.8600	1.73211	0.866055	−	0.499949i	$-0.166648\pi$
$701$	45.8600	1.73211	0.866055	+	0.499949i	$0.166648\pi$
$702$	0	0
$703$	−59.8653	−2.25786
$704$	0	0
$705$	−7.05137	−0.265570
$706$	0	0
$707$	−0.108027	−0.00406278
$708$	0	0
$709$	−15.0380	−0.564766	−0.282383	−	0.959302i	$-0.591125\pi$
$709$	−15.0380	−0.564766	−0.282383	+	0.959302i	$0.591125\pi$
$710$	0	0
$711$	0.156762	0.00587904
$712$	0	0
$713$	−6.36226	−0.238268
$714$	0	0
$715$	15.9327	0.595847
$716$	0	0
$717$	−29.3463	−1.09596
$718$	0	0
$719$	15.6244	0.582692	0.291346	−	0.956618i	$-0.405897\pi$
$719$	15.6244	0.582692	0.291346	+	0.956618i	$0.405897\pi$
$720$	0	0
$721$	−1.05665	−0.0393518
$722$	0	0
$723$	13.9327	0.518161
$724$	0	0
$725$	7.96633	0.295862
$726$	0	0
$727$	−8.77152	−0.325318	−0.162659	−	0.986682i	$-0.552007\pi$
$727$	−8.77152	−0.325318	−0.162659	+	0.986682i	$0.552007\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.1027	0.521609
$732$	0	0
$733$	4.75646	0.175684	0.0878419	−	0.996134i	$-0.472003\pi$
$733$	4.75646	0.175684	0.0878419	+	0.996134i	$0.472003\pi$
$734$	0	0
$735$	−6.84324	−0.252417
$736$	0	0
$737$	38.5544	1.42017
$738$	0	0
$739$	−17.8078	−0.655072	−0.327536	−	0.944839i	$-0.606218\pi$
$739$	−17.8078	−0.655072	−0.327536	+	0.944839i	$0.606218\pi$
$740$	0	0
$741$	−34.8273	−1.27941
$742$	0	0
$743$	−40.3703	−1.48104	−0.740522	−	0.672033i	$-0.765421\pi$
$743$	−40.3703	−1.48104	−0.740522	+	0.672033i	$0.765421\pi$
$744$	0	0
$745$	8.79186	0.322109
$746$	0	0
$747$	−7.74049	−0.283210
$748$	0	0
$749$	4.06207	0.148425
$750$	0	0
$751$	45.3817	1.65600	0.828001	−	0.560727i	$-0.189478\pi$
$751$	45.3817	1.65600	0.828001	+	0.560727i	$0.189478\pi$
$752$	0	0
$753$	−7.68648	−0.280111
$754$	0	0
$755$	−23.7759	−0.865293
$756$	0	0
$757$	13.9390	0.506621	0.253310	−	0.967385i	$-0.418481\pi$
$757$	13.9390	0.506621	0.253310	+	0.967385i	$0.418481\pi$
$758$	0	0
$759$	17.7626	0.644740
$760$	0	0
$761$	−46.5067	−1.68587	−0.842934	−	0.538017i	$-0.819174\pi$
$761$	−46.5067	−1.68587	−0.842934	+	0.538017i	$0.819174\pi$
$762$	0	0
$763$	5.37295	0.194514
$764$	0	0
$765$	−4.25951	−0.154003
$766$	0	0
$767$	−19.0868	−0.689184
$768$	0	0
$769$	−7.44098	−0.268329	−0.134164	−	0.990959i	$-0.542835\pi$
$769$	−7.44098	−0.268329	−0.134164	+	0.990959i	$0.542835\pi$
$770$	0	0
$771$	13.3596	0.481135
$772$	0	0
$773$	−44.9027	−1.61504	−0.807518	−	0.589842i	$-0.799190\pi$
$773$	−44.9027	−1.61504	−0.807518	+	0.589842i	$0.799190\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−3.88392	−0.139335
$778$	0	0
$779$	19.5784	0.701471
$780$	0	0
$781$	37.8299	1.35366
$782$	0	0
$783$	−7.96633	−0.284693
$784$	0	0
$785$	−17.5164	−0.625186
$786$	0	0
$787$	9.45431	0.337010	0.168505	−	0.985701i	$-0.446106\pi$
$787$	9.45431	0.337010	0.168505	+	0.985701i	$0.446106\pi$
$788$	0	0
$789$	13.8299	0.492358
$790$	0	0
$791$	3.93265	0.139829
$792$	0	0
$793$	72.0301	2.55786
$794$	0	0
$795$	−1.05137	−0.0372884
$796$	0	0
$797$	44.4650	1.57503	0.787516	−	0.616295i	$-0.211367\pi$
$797$	44.4650	1.57503	0.787516	+	0.616295i	$0.211367\pi$
$798$	0	0
$799$	−30.0354	−1.06258
$800$	0	0
$801$	7.55005	0.266768
$802$	0	0
$803$	−38.5544	−1.36056
$804$	0	0
$805$	2.51902	0.0887838
$806$	0	0
$807$	−13.0044	−0.457775
$808$	0	0
$809$	38.1311	1.34062	0.670310	−	0.742081i	$-0.266161\pi$
$809$	38.1311	1.34062	0.670310	+	0.742081i	$0.266161\pi$
$810$	0	0
$811$	−27.4136	−0.962623	−0.481311	−	0.876550i	$-0.659839\pi$
$811$	−27.4136	−0.962623	−0.481311	+	0.876550i	$0.659839\pi$
$812$	0	0
$813$	7.61913	0.267215
$814$	0	0
$815$	−16.4987	−0.577924
$816$	0	0
$817$	20.2055	0.706901
$818$	0	0
$819$	−2.25951	−0.0789537
$820$	0	0
$821$	−2.96369	−0.103433	−0.0517167	−	0.998662i	$-0.516469\pi$
$821$	−2.96369	−0.103433	−0.0517167	+	0.998662i	$0.516469\pi$
$822$	0	0
$823$	1.20814	0.0421130	0.0210565	−	0.999778i	$-0.493297\pi$
$823$	1.20814	0.0421130	0.0210565	+	0.999778i	$0.493297\pi$
$824$	0	0
$825$	2.79186	0.0972002
$826$	0	0
$827$	−2.70245	−0.0939733	−0.0469867	−	0.998896i	$-0.514962\pi$
$827$	−2.70245	−0.0939733	−0.0469867	+	0.998896i	$0.514962\pi$
$828$	0	0
$829$	28.5544	0.991736	0.495868	−	0.868398i	$-0.334850\pi$
$829$	28.5544	0.991736	0.495868	+	0.868398i	$0.334850\pi$
$830$	0	0
$831$	17.2232	0.597466
$832$	0	0
$833$	−29.1488	−1.00995
$834$	0	0
$835$	−10.8946	−0.377024
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	43.7289	1.50969	0.754844	−	0.655904i	$-0.227712\pi$
$839$	43.7289	1.50969	0.754844	+	0.655904i	$0.227712\pi$
$840$	0	0
$841$	34.4624	1.18836
$842$	0	0
$843$	17.6191	0.606835
$844$	0	0
$845$	−19.5678	−0.673151
$846$	0	0
$847$	1.26916	0.0436088
$848$	0	0
$849$	8.08241	0.277388
$850$	0	0
$851$	−62.4110	−2.13942
$852$	0	0
$853$	−44.9300	−1.53837	−0.769187	−	0.639024i	$-0.779339\pi$
$853$	−44.9300	−1.53837	−0.769187	+	0.639024i	$0.779339\pi$
$854$	0	0
$855$	−6.10275	−0.208710
$856$	0	0
$857$	−16.8946	−0.577109	−0.288554	−	0.957463i	$-0.593175\pi$
$857$	−16.8946	−0.577109	−0.288554	+	0.957463i	$0.593175\pi$
$858$	0	0
$859$	−35.9186	−1.22553	−0.612764	−	0.790266i	$-0.709942\pi$
$859$	−35.9186	−1.22553	−0.612764	+	0.790266i	$0.709942\pi$
$860$	0	0
$861$	1.27020	0.0432884
$862$	0	0
$863$	48.0708	1.63635	0.818175	−	0.574970i	$-0.194986\pi$
$863$	48.0708	1.63635	0.818175	+	0.574970i	$0.194986\pi$
$864$	0	0
$865$	16.8946	0.574434
$866$	0	0
$867$	−1.14343	−0.0388329
$868$	0	0
$869$	−0.437658	−0.0148465
$870$	0	0
$871$	−78.8087	−2.67033
$872$	0	0
$873$	4.79186	0.162180
$874$	0	0
$875$	0.395932	0.0133849
$876$	0	0
$877$	28.6979	0.969058	0.484529	−	0.874775i	$-0.338991\pi$
$877$	28.6979	0.969058	0.484529	+	0.874775i	$0.338991\pi$
$878$	0	0
$879$	−29.6731	−1.00085
$880$	0	0
$881$	25.8583	0.871188	0.435594	−	0.900143i	$-0.356538\pi$
$881$	25.8583	0.871188	0.435594	+	0.900143i	$0.356538\pi$
$882$	0	0
$883$	17.0328	0.573198	0.286599	−	0.958051i	$-0.407475\pi$
$883$	17.0328	0.573198	0.286599	+	0.958051i	$0.407475\pi$
$884$	0	0
$885$	−3.34456	−0.112426
$886$	0	0
$887$	7.23021	0.242767	0.121383	−	0.992606i	$-0.461267\pi$
$887$	7.23021	0.242767	0.121383	+	0.992606i	$0.461267\pi$
$888$	0	0
$889$	−7.05943	−0.236766
$890$	0	0
$891$	−2.79186	−0.0935310
$892$	0	0
$893$	−43.0328	−1.44004
$894$	0	0
$895$	2.93529	0.0981160
$896$	0	0
$897$	−36.3082	−1.21230
$898$	0	0
$899$	−7.96633	−0.265692
$900$	0	0
$901$	−4.47834	−0.149195
$902$	0	0
$903$	1.31088	0.0436235
$904$	0	0
$905$	−7.03804	−0.233952
$906$	0	0
$907$	15.2499	0.506363	0.253182	−	0.967419i	$-0.418523\pi$
$907$	15.2499	0.506363	0.253182	+	0.967419i	$0.418523\pi$
$908$	0	0
$909$	0.272843	0.00904962
$910$	0	0
$911$	30.4517	1.00891	0.504455	−	0.863438i	$-0.331694\pi$
$911$	30.4517	1.00891	0.504455	+	0.863438i	$0.331694\pi$
$912$	0	0
$913$	21.6104	0.715200
$914$	0	0
$915$	12.6218	0.417263
$916$	0	0
$917$	−5.92460	−0.195648
$918$	0	0
$919$	−29.2435	−0.964655	−0.482328	−	0.875991i	$-0.660209\pi$
$919$	−29.2435	−0.964655	−0.482328	+	0.875991i	$0.660209\pi$
$920$	0	0
$921$	−13.9797	−0.460645
$922$	0	0
$923$	−77.3277	−2.54527
$924$	0	0
$925$	−9.80957	−0.322537
$926$	0	0
$927$	2.66877	0.0876541
$928$	0	0
$929$	−11.8282	−0.388070	−0.194035	−	0.980995i	$-0.562158\pi$
$929$	−11.8282	−0.388070	−0.194035	+	0.980995i	$0.562158\pi$
$930$	0	0
$931$	−41.7626	−1.36871
$932$	0	0
$933$	32.4180	1.06132
$934$	0	0
$935$	11.8920	0.388909
$936$	0	0
$937$	−20.6572	−0.674840	−0.337420	−	0.941354i	$-0.609554\pi$
$937$	−20.6572	−0.674840	−0.337420	+	0.941354i	$0.609554\pi$
$938$	0	0
$939$	22.1178	0.721787
$940$	0	0
$941$	−48.0284	−1.56568	−0.782840	−	0.622222i	$-0.786230\pi$
$941$	−48.0284	−1.56568	−0.782840	+	0.622222i	$0.786230\pi$
$942$	0	0
$943$	20.4110	0.664673
$944$	0	0
$945$	−0.395932	−0.0128797
$946$	0	0
$947$	25.3242	0.822926	0.411463	−	0.911426i	$-0.365018\pi$
$947$	25.3242	0.822926	0.411463	+	0.911426i	$0.365018\pi$
$948$	0	0
$949$	78.8087	2.55824
$950$	0	0
$951$	−25.3596	−0.822342
$952$	0	0
$953$	−37.5030	−1.21484	−0.607421	−	0.794380i	$-0.707796\pi$
$953$	−37.5030	−1.21484	−0.607421	+	0.794380i	$0.707796\pi$
$954$	0	0
$955$	12.8609	0.416170
$956$	0	0
$957$	22.2409	0.718946
$958$	0	0
$959$	−5.89197	−0.190262
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−10.2595	−0.330608
$964$	0	0
$965$	−23.9327	−0.770419
$966$	0	0
$967$	−26.1788	−0.841855	−0.420927	−	0.907094i	$-0.638295\pi$
$967$	−26.1788	−0.841855	−0.420927	+	0.907094i	$0.638295\pi$
$968$	0	0
$969$	−25.9947	−0.835071
$970$	0	0
$971$	27.8316	0.893160	0.446580	−	0.894744i	$-0.352642\pi$
$971$	27.8316	0.893160	0.446580	+	0.894744i	$0.352642\pi$
$972$	0	0
$973$	−4.51902	−0.144873
$974$	0	0
$975$	−5.70682	−0.182764
$976$	0	0
$977$	−11.0380	−0.353138	−0.176569	−	0.984288i	$-0.556500\pi$
$977$	−11.0380	−0.353138	−0.176569	+	0.984288i	$0.556500\pi$
$978$	0	0
$979$	−21.0787	−0.673679
$980$	0	0
$981$	−13.5704	−0.433269
$982$	0	0
$983$	2.10275	0.0670673	0.0335336	−	0.999438i	$-0.489324\pi$
$983$	2.10275	0.0670673	0.0335336	+	0.999438i	$0.489324\pi$
$984$	0	0
$985$	19.0868	0.608155
$986$	0	0
$987$	−2.79186	−0.0888660
$988$	0	0
$989$	21.0647	0.669819
$990$	0	0
$991$	−13.1321	−0.417153	−0.208577	−	0.978006i	$-0.566883\pi$
$991$	−13.1321	−0.417153	−0.208577	+	0.978006i	$0.566883\pi$
$992$	0	0
$993$	25.9460	0.823371
$994$	0	0
$995$	−14.5324	−0.460707
$996$	0	0
$997$	36.3489	1.15118	0.575591	−	0.817738i	$-0.304772\pi$
$997$	36.3489	1.15118	0.575591	+	0.817738i	$0.304772\pi$
$998$	0	0
$999$	9.80957	0.310361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bn.1.2		3
4.3	odd	2		1860.2.a.g.1.2	✓	3
12.11	even	2		5580.2.a.j.1.2		3
20.3	even	4		9300.2.g.q.3349.5		6
20.7	even	4		9300.2.g.q.3349.2		6
20.19	odd	2		9300.2.a.u.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.2	✓	3	4.3	odd	2
5580.2.a.j.1.2		3	12.11	even	2
7440.2.a.bn.1.2		3	1.1	even	1	trivial
9300.2.a.u.1.2		3	20.19	odd	2
9300.2.g.q.3349.2		6	20.7	even	4
9300.2.g.q.3349.5		6	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -0.395932 q^{7} +1.00000 q^{9} -2.79186 q^{11} +5.70682 q^{13} +1.00000 q^{15} +4.25951 q^{17} +6.10275 q^{19} +0.395932 q^{21} +6.36226 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.96633 q^{29} -1.00000 q^{31} +2.79186 q^{33} +0.395932 q^{35} -9.80957 q^{37} -5.70682 q^{39} +3.20814 q^{41} +3.31088 q^{43} -1.00000 q^{45} -7.05137 q^{47} -6.84324 q^{49} -4.25951 q^{51} -1.05137 q^{53} +2.79186 q^{55} -6.10275 q^{57} -3.34456 q^{59} +12.6218 q^{61} -0.395932 q^{63} -5.70682 q^{65} -13.8096 q^{67} -6.36226 q^{69} -13.5501 q^{71} +13.8096 q^{73} -1.00000 q^{75} +1.10539 q^{77} +0.156762 q^{79} +1.00000 q^{81} -7.74049 q^{83} -4.25951 q^{85} -7.96633 q^{87} +7.55005 q^{89} -2.25951 q^{91} +1.00000 q^{93} -6.10275 q^{95} +4.79186 q^{97} -2.79186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{15} + 10 q^{17} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 16 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 3 * q^15 + 10 * q^17 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 16 * q^41 - 2 * q^43 - 3 * q^45 - 12 * q^47 - 5 * q^49 - 10 * q^51 + 6 * q^53 + 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^63 - 2 * q^65 - 8 * q^67 + 2 * q^69 - 10 * q^71 + 8 * q^73 - 3 * q^75 + 28 * q^77 + 16 * q^79 + 3 * q^81 - 26 * q^83 - 10 * q^85 - 6 * q^87 - 8 * q^89 - 4 * q^91 + 3 * q^93 + 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more